In this example, we will solve the 1D advection equation
$$\frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = 0,$$ where $a = 1$ is a constant.
using the initial condition
$$u(x,0) = \begin{cases} \sin^4 2\pi x & 0 \le x \le \frac12 \\ 1 & \frac46 \le x \le \frac56 \\ 0 & \text{otherwise} \end{cases}$$We also use periodic boundary conditions $u(0,t) = u(1,t)$ for all $t \ge 0$. This allows us to understand how each solver affects the solution in terms of smoothing and phase shift.
Below, we compare four different hyperbolic solvers: